Assessing a Local Ensemble Kalman Filter

Istvan Szunyogh“Chaos-Weather Team”

University of Maryland College Park

IPAM DA Workshop, UCLA, February 22-25, 2005

System Components• Data assimilation scheme: Local Ensemble

(Transform) Kalman Filter (Ott et al. 2002, 2004; Hunt 2005) implemented by Eric Kostelich (ASU) and I. Sz.

• Model: Operational Global Forecast System (GFS) of the National Centers for Environmental Prediction/National Weather Service

• Model Resolution: T62 (~150 km) in the horizontal directions and 28 vertical level dimension of the state vector:1,137,024; dimension of the grid space (analysis space): 2,544,768]

WARNING!!!!!

All results shown in this presentation were obtained for

the perfect model scenario

Why a Perfect Model?

• Easier to find bugs in the code• To expose weaknesses of the scheme

(model errors cannot be blamed for unexpected bad results)

• To establish a reference needed to assess the effects of model errors

• To learn more about the dynamics of the model (predictability, dimensionality, etc.)

Local Ensemble Kalman FilterIllustration on a two dimensional grid

• The state estimate is updated at the center grid point

• The background state is considered only from a local region (yellow dots)

• All observations are considered from the local region (purple diamonds)

Base Experiment

• Number of ensemble members: 40 • Local regions: 7x7xV grid point cubes; V=1, 3, 5,

7 • Variance Inflation: Multiplicative, uniform 4%

(needed to compensate for the loss of variance due to nonlinearities and sampling errors)

• Observations: 2000 vertical sounding of wind, temperature, and surface pressure

Depth of Local Cubes

Mid-troposphere

Lower troposphere

Upper troposphere

Lower stratosphere

Dimension of Local State Vector ~1,700

Time evolution of errors

analysis cycle (time)

Rmsanalysiserror

surface pressure

Observational error

The error settles at a similarly rapid speed for all variables15-days (60 cycles) is a safe upper bound estimate for the transient

Zonal-Mean Analysis Error (45-day mean) The analysis errors are much smaller than the observational

errorsTemperature u-wind

The “largest” errors: deep convection (maximum CAPE), polar regions

Time-Mean Analysis Error45-day average

Temperature 60 kPa u-wind 30 kPa

The figures confirm the conclusions drawn based on zonal means

N-America

S-AmericaAustralia

Africa

Euro-Asia

Tropics

SH Extratropics

NH Extratropics

E-dimensionA local measure of complexity

Illustration in 2D model grid space

1 Number of EnsembleMembers-1

E-dimension

Complexity:

Based on the eigenvalues

A spatio-temporallychanging scalar valueis assigned to each grid point

Introduced byPatil, Hunt et al. (2001)Studied in details byOczkowski et al (2005)

€

σiof the ensemble based estimate of the local covariancematrix:

€

€

2

iσi∑ ⎛ ⎝ ⎜ ⎞

⎠ ⎟ iσ 2

i∑

The more unevenly distributed the variance inthe ensemble space, the lower the E-dimension

Explained (Background) Error Variance

Illustration for a rank-2 covariance matrix (3-member ensemble)

Background mean

True state

Eigenvector 1

Eigenvector 2

bbe

Explained Variance: be2/ b2

Projection on the plane of theeigenvectors

A perfect explained variance of 1 implies that the space of uncertainties iscorrectly captured by the ensemble, but it does not guarantee that the distribution of the variance within that space is correctly represented by the ensemble

E-dimension, Explained Variance, Analysis Error

• Background ensemble perturbations span the space, where corrections to the estimate of the state can be made => E-dimension characterizes the distribution of variance between distinct state space directions, EV measures the potential for a correction

• A correction is realized, when the difference between the observation and the background mean has a projection on the subspace that contributes to the explained variance

• Low explained variance or low E-dimension would be a problem if the error in the resulting state estimate was large

E-Dimension and Explained Variance

E-dimension Explained BackgroundVariance

The E-dimension and Explained Background Variance seem to be stronglyanti-correlated. The E-dimensions is the highest, the explained variance is the lowest, in the Tropics. Szunyogh et al. (2005)

E-Dimension vs. Explained Variance

Zonal Mean No Averaging

E-di

men

sion

Explained Variance Explained Variance

E-di

men

sion

Correlation:-0.91 Correlation:-0.9

Low E-dimension always indicates high explained variance

High E-dimension always indicates low explained varianceSzunyogh et al. (2005)

Sensitivity to the Size of the Local Region: Part I

Temperature, Global Error

rms error

3x3xV5x5xV7x7xV9x9xV11x11xV

Observationalerror

The performance is onlymodestly sensitive to thelocal region size

bestworst

mid-troposphere

Szunyogh et al. (2005)

Sensitivity to the Size of the Local Region: Part II

u-wind, NH extra-tropics u-wind, Tropics

The tropical windis the most sensitiveanalysis variable

Best: 5x5xV

Worst:11x11xV

Observational errorSzunyogh et al. (2005)

Sensitivity to the Size of the Local Region and the Ensemble

Size u-wind, Tropics, 80-member ensemble

Observational error

The 7x7xV localizationbreaks even with the5x5xV localization

the5x5xV localization improves only a little withincreasing the ensemble size

Szunyogh et al. (2005)

Sensitivity to the Number of Observations

500 soundings1000 soundings2000 soundings18048 soundings(All locations)

Global Temperature Error Wind Error in Tropics

Szunyogh et al. (2005)

E-dimension and Explained Variance (fully observed atmosphere)

E-dimension Explained Variance

The largest E-dimension did not changeThe smallest explained variancewas reduced by about 0.05 (about 12%)

Error reduction in the tropics isabout 46%

Szunyogh et al. (2005)

Evolution of the Forecast Errors

45-day mean

As forecast timeincreases the extratropicalstorm track regions becomethe regions oflargest error

D. Kuhl et al.

Evolution of the E-dimension

The E-dimension rapidlyDecreases in the stormTrack regions

The error growth andthe decrease of theE-dimension is closelyrelated

D. Kuhl et al.

Evolution of the Explained Variance

The explained variance isthe largest in the storm trackregions and it increases withtime

Large error growth, low E-dimension, and large explained varianceare closely related

There seems to exist a ‘localanalogue’ to the unstable subspace

D. Kuhl et al.

E-d

imen

sion

Explained Variance

The scatter plots confirmthe increasingly closecorrespondence betweenlow E-dimensionality andhigh explained variance(improving ensemble performance)

D. Kuhl et al.

Time Mean Evolution of the Forecast Errors

0

0.5

1

1.5

2

2.5

3

3.5

4

0 12 24 36 48 60 72 84 96 108 120 132

Forecast (Hour)

Average Forecast Error

Extra Trop. NHExtra Trop. SHTropics (linear growth)

(exponential growth)

Curves fitted forFirst 72 hours

The error doubling time in the extratropics is about35-37 hours

D. Kuhl et al.

The effect of Local Patch Size on the Error Growth in the NH Extratropics…

0.1

1

10

0 12 24 36 48 60 72 84

Forecast (Hour)

Average Forecast Error

Patch Size 9x9Patch Size 7x7Patch Size 5x5

is negligible

Forecast hourD. Kuhl et al.

The Effect of the Ensemble Size on the Forecast Errors in the NH Extratropics…

0.1

1

10

0 12 24 36 48 60 72 84Forecast (Hour)

Average Forecast Error

40 Mem. Ensemble80 Mem. Ensemble

is negligible

Forecast hourD. Kuhl et al.

The Effect of the Number of Observations on the Forecast Errors in the NH Extratropics

0.1

1

10

0 12 24 36 48 60 72 84

Forecast (Hour)

Average Forecast Error

500 Observations1,000 Observations2000 Observations17,848 Observations

Forecast hour

The slightly larger growth rate for the initially smaller errors indicates the presence of saturation processes

D. Kuhl et al.

Conclusions and Challenges• The state-of-the art model shows local low-dimensional

behavior. Is it reasonable to assume that the real atmosphere shows a similar behavior? (My guess: yes)

• Local low dimensionality helps obtain more accurate estimate of the initial state and more accurate prediction of the forecast uncertainties.

• Localization in the physical space seems to be a practical way to apply low dimensional concepts to a very high dimensional system. Is it possible to develop a rigorous theoretical framework to support this phenomenological result? (I have no guess)

• On the practical side, the LETKF assimilates an operational observation file (excluding satellite radiances) in 5 minutes

References• Kuhl, D., I. Szunyogh, E. J. Kostelich, G. Gyarmati, D.J. Patil, M. Oczkowski, B. Hunt,

E. Kalnay, E. Ott, J. A. Yorke, 2005: Assessing predictability with a Local Ensemble Kalman Filter (to be submitted)

• Szunyogh, I, E. J. Kostelich, G. Gyarmati, D. J. Patil, B. R. Hunt, E. Kalnay, E. Ott, and J. A. Yorke, 2005: Assessing a local ensemble Kalman filter: Perfect model experiments with the NCEP global model. Tellus 57A. [in print]

• Oczkowski, M., I. Szunyogh, and D. J. Patil, 2005: Mechanisms for the development of locally low dimensional atmospheric dynamics. J. Atmos. Sci. [in print].

• Ott, E., B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich, M. Corazza, E. Kalnay, D. J. Patil, J. A. Yorke, 2004: A local ensemble Kalman Filter for atmospheric data assimilation.Tellus 56A , 415-428.

• Ott, E., B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich, M. Corazza, E. Kalnay, D. J. Patil, and J. A. Yorke, 2004: Estimating the state of large spatio-temporally chaotic systems. Phys. Lett. A., 330, 365-370.

• Patil, D. J., B. R. Hunt, E. Kalnay, J. A. Yorke, and E. Ott, 2001: Local low dimensionality of atmospheric dynamics, Phys. Rev. Let., 86, 5878-5881.

• Reprints and preprints of papers by our group are available at http://keck2.umd.edu/weather/weather_publications.htm